Precalculus Examples

$\frac{3 \sqrt{60}}{2 \sqrt{27}}$

Step 1

Combine $\sqrt{60}$ and $\sqrt{27}$ into a single radical.

$\frac{3 \sqrt{\frac{60}{27}}}{2}$

Step 2

Cancel the common factor of $60$ and $27$ .

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Step 2.1

Factor $3$ out of $60$ .

$\frac{3 \sqrt{\frac{3 (20)}{27}}}{2}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $27$ .

$\frac{3 \sqrt{\frac{3 \cdot 20}{3 \cdot 9}}}{2}$

Step 2.2.2

Cancel the common factor.

$\frac{3 \sqrt{\frac{3 \cdot 20}{3 \cdot 9}}}{2}$

Step 2.2.3

Rewrite the expression.

$\frac{3 \sqrt{\frac{20}{9}}}{2}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $\sqrt{\frac{20}{9}}$ as $\frac{\sqrt{20}}{\sqrt{9}}$ .

$\frac{3 \frac{\sqrt{20}}{\sqrt{9}}}{2}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 3.2.1.1

Factor $4$ out of $20$ .

$\frac{3 \frac{\sqrt{4 (5)}}{\sqrt{9}}}{2}$

Step 3.2.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{3 \frac{\sqrt{2^{2} \cdot 5}}{\sqrt{9}}}{2}$

Step 3.2.2

Pull terms out from under the radical.

$\frac{3 \frac{2 \sqrt{5}}{\sqrt{9}}}{2}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Rewrite $9$ as $3^{2}$ .

$\frac{3 \frac{2 \sqrt{5}}{\sqrt{3^{2}}}}{2}$

Step 3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 \frac{2 \sqrt{5}}{3}}{2}$

Step 4

Combine $3$ and $\frac{2 \sqrt{5}}{3}$ .

$\frac{\frac{3 (2 \sqrt{5})}{3}}{2}$

Step 5

Multiply $3$ by $2$ .

$\frac{\frac{6 \sqrt{5}}{3}}{2}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Reduce the expression $\frac{6 \sqrt{5}}{3}$ by cancelling the common factors.

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Step 6.1.1

Factor $3$ out of $6 \sqrt{5}$ .

$\frac{\frac{3 (2 \sqrt{5})}{3}}{2}$

Step 6.1.2

Factor $3$ out of $3$ .

$\frac{\frac{3 (2 \sqrt{5})}{3 (1)}}{2}$

Step 6.1.3

Cancel the common factor.

$\frac{\frac{3 (2 \sqrt{5})}{3 \cdot 1}}{2}$

Step 6.1.4

Rewrite the expression.

$\frac{\frac{2 \sqrt{5}}{1}}{2}$

Step 6.2

Divide $2 \sqrt{5}$ by $1$ .

$\frac{2 \sqrt{5}}{2}$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Cancel the common factor.

$\frac{2 \sqrt{5}}{2}$

Step 7.2

Divide $\sqrt{5}$ by $1$ .

$\sqrt{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt{5}$

Decimal Form:

$2.23606797 \dots$